Grades 11-12 Video Solutions 2010-11&12 Video Solutions 2010 problem9

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Question number 10. A certain 2-digit number multiplied by 2 and written in front of the  original 2-digit number gives a 4-digit number divisible by 30. How many such 4-digit numbers are there?  We can begin by writing down here our 4-digit example where we have digits A, B, C and D and what we know is that here divisibility by 30 also tells us that we have divisibility  by 10. So, D is in fact 0 and then reading the problem again, the 2-digit number which is CD here is multiplied by 2 and written in front of the original 2-digit number and  what we write in front is the AB. So, what we know in addition is that C here is an element of the set containing 1, 2, 3, 4 and so on all the way up to 9 and then we  just test what happens. If C is equal to 1, then 2C is equal to 2 and the number we are looking at is 2010. Now, we notice that we can check everything works. The original  number was 10. We multiply that by 2 and put it in front of 10 and we have 20 followed by 10. So, that's one such number. If C is equal to 2, what we have is 2C is equal to  4 and then the original number was 20 and in front of that we have a 40. So, 4020. If C is equal to 3, 2C is 6 and then we can write down our number. That would be 60 followed  by 30. If C is equal to 4, then 2C is 8 and what we have is 80 followed by 40 and then we have to stop. So, these are the only such numbers because if C is increased to 5, what  we have is the number 50 here and then in front of it we would have to stick in double 50 so 100 and that is now 5 digits. So, that's not possible. We stop with our 4 cases and  so there are 4 such numbers.

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